3.4.99 \(\int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [399]

Optimal. Leaf size=238 \[ -\frac {\sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {25 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \]

[Out]

25/8*arctanh((1+tan(f*x+e))^(1/2))/f-arctan((3-2*2^(1/2)+(1-2^(1/2))*tan(f*x+e))/(-14+10*2^(1/2))^(1/2)/(1+tan
(f*x+e))^(1/2))*(2^(1/2)-1)^(1/2)/f-arctanh((3+2*2^(1/2)+(1+2^(1/2))*tan(f*x+e))/(14+10*2^(1/2))^(1/2)/(1+tan(
f*x+e))^(1/2))*(1+2^(1/2))^(1/2)/f+7/8*cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f-7/12*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2
)/f-1/3*cot(f*x+e)^3*(1+tan(f*x+e))^(1/2)/f

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Rubi [A]
time = 0.36, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3648, 3731, 3730, 3735, 12, 3617, 3616, 209, 213, 3715, 65} \begin {gather*} -\frac {\sqrt {\sqrt {2}-1} \text {ArcTan}\left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{f}+\frac {25 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]

[Out]

-((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + T
an[e + f*x]])])/f) + (25*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] +
(1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f + (7*Cot[e + f*x]*Sqrt[1 + Ta
n[e + f*x]])/(8*f) - (7*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]]
)/(3*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 3616

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(
d^2/f), Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]
]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2
*a*c*d - b*(c^2 - d^2), 0]

Rule 3617

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> With[{q =
Rt[a^2 + b^2, 2]}, Dist[1/(2*q), Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*
x]], x], x] - Dist[1/(2*q), Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x
], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2
*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])

Rule 3648

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[
1/((m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[a*c^2*(m + 1) + a*
d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d
^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[2*m]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3731

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*t
an[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x]
)^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[
e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1)
+ a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^
2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3735

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + a^2*C)/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^
2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -1]

Rubi steps

\begin {align*} \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx &=-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{3} \int \frac {\cot ^3(e+f x) \left (-\frac {7}{2}+\frac {5}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {1}{6} \int \frac {\cot ^2(e+f x) \left (-\frac {21}{4}-12 \tan (e+f x)-\frac {21}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{6} \int \frac {\cot (e+f x) \left (\frac {75}{8}-\frac {21}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{6} \int -\frac {12 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx-\frac {25}{16} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+2 \int \frac {\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx-\frac {25 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {\int \frac {1+\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{\sqrt {2}}+\frac {\int \frac {1+\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{\sqrt {2}}-\frac {25 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{8 f}\\ &=\frac {25 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {\left (4-3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}\\ &=-\frac {\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {25 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 19.14, size = 321, normalized size = 1.35 \begin {gather*} \frac {\cos (2 (e+f x)) \csc ^4(e+f x) \sec (e+f x) \left (34 \cos (e+f x)+30 \cos (3 (e+f x))-82 \sin (e+f x)-192 \sqrt {-2-2 i} \text {ArcTan}\left (\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}\right ) \sin ^3(e+f x) \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}-192 \sqrt {-2+2 i} \text {ArcTan}\left (\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}\right ) \sin ^3(e+f x) \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}-600 \tanh ^{-1}\left (\sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}\right ) \sin ^3(e+f x) \sqrt {1+\sec (e+f x) \sqrt {\sin ^2(e+f x)}}+86 \sin (3 (e+f x))\right ) \sqrt {1+\tan (e+f x)}}{192 f (1+\cot (e+f x)) \left (-2+\sec ^2(e+f x)\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]

[Out]

(Cos[2*(e + f*x)]*Csc[e + f*x]^4*Sec[e + f*x]*(34*Cos[e + f*x] + 30*Cos[3*(e + f*x)] - 82*Sin[e + f*x] - 192*S
qrt[-2 - 2*I]*ArcTan[Sqrt[-1/2 - I/2]*Sqrt[1 + Sec[e + f*x]*Sqrt[Sin[e + f*x]^2]]]*Sin[e + f*x]^3*Sqrt[1 + Sec
[e + f*x]*Sqrt[Sin[e + f*x]^2]] - 192*Sqrt[-2 + 2*I]*ArcTan[Sqrt[-1/2 + I/2]*Sqrt[1 + Sec[e + f*x]*Sqrt[Sin[e
+ f*x]^2]]]*Sin[e + f*x]^3*Sqrt[1 + Sec[e + f*x]*Sqrt[Sin[e + f*x]^2]] - 600*ArcTanh[Sqrt[1 + Sec[e + f*x]*Sqr
t[Sin[e + f*x]^2]]]*Sin[e + f*x]^3*Sqrt[1 + Sec[e + f*x]*Sqrt[Sin[e + f*x]^2]] + 86*Sin[3*(e + f*x)])*Sqrt[1 +
Tan[e + f*x]])/(192*f*(1 + Cot[e + f*x])*(-2 + Sec[e + f*x]^2))

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.74, size = 11278, normalized size = 47.39

method	result	size
default	\(\text {Expression too large to display}\)	\(11278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((tan(f*x + e) + 1)^(3/2)*cot(f*x + e)^4, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1343 vs. \(2 (200) = 400\).
time = 1.33, size = 1343, normalized size = 5.64 \begin {gather*} -\frac {6 \cdot 8^{\frac {1}{4}} {\left (2 \, f \cos \left (f x + e\right )^{4} - 4 \, f \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{4} - 2 \, f^{3} \cos \left (f x + e\right )^{2} + f^{3}\right )} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) - 6 \cdot 8^{\frac {1}{4}} {\left (2 \, f \cos \left (f x + e\right )^{4} - 4 \, f \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{4} - 2 \, f^{3} \cos \left (f x + e\right )^{2} + f^{3}\right )} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) - 75 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + 1\right ) + 75 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - 1\right ) - 2 \, {\left (14 \, \cos \left (f x + e\right )^{4} - 14 \, \cos \left (f x + e\right )^{2} - {\left (29 \, \cos \left (f x + e\right )^{3} - 21 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + \frac {24 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (f^{5} \cos \left (f x + e\right )^{4} - 2 \, f^{5} \cos \left (f x + e\right )^{2} + f^{5}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{16} \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - \frac {1}{8} \cdot 8^{\frac {3}{4}} {\left (2 \, f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right )}{f^{4}} + \frac {24 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (f^{5} \cos \left (f x + e\right )^{4} - 2 \, f^{5} \cos \left (f x + e\right )^{2} + f^{5}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{16} \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - \frac {1}{8} \cdot 8^{\frac {3}{4}} {\left (2 \, f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right )}{f^{4}}}{48 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/48*(6*8^(1/4)*(2*f*cos(f*x + e)^4 - 4*f*cos(f*x + e)^2 + sqrt(2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos(f*x + e)^2
+ f^3)*sqrt(f^(-4)) + 2*f)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(2*(2*sqrt(2)*f^2*sqrt(f^(
-4))*cos(f*x + e) + 8^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt(
f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e)
)/cos(f*x + e)) - 6*8^(1/4)*(2*f*cos(f*x + e)^4 - 4*f*cos(f*x + e)^2 + sqrt(2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos
(f*x + e)^2 + f^3)*sqrt(f^(-4)) + 2*f)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(2*(2*sqrt(2)*f
^2*sqrt(f^(-4))*cos(f*x + e) - 8^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-2*sqrt(2
)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*s
in(f*x + e))/cos(f*x + e)) - 75*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*log(sqrt((cos(f*x + e) + sin(f*x + e))
/cos(f*x + e)) + 1) + 75*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*
x + e)) - 1) - 2*(14*cos(f*x + e)^4 - 14*cos(f*x + e)^2 - (29*cos(f*x + e)^3 - 21*cos(f*x + e))*sin(f*x + e))*
sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 24*8^(1/4)*sqrt(2)*(f^5*cos(f*x + e)^4 - 2*f^5*cos(f*x + e)
^2 + f^5)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*arctan(1/16*8^(3/4)*sqrt(2)*(2*f^5*sqrt(f^(-4))
+ sqrt(2)*f^3)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 8^(1/4)*
(sqrt(2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x +
e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/
4) - 1/8*8^(3/4)*(2*f^5*sqrt(f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) +
sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - f^2*sqrt(f^(-4)) - sqrt(2))/f^4 + 24*8^(1/4)*sqrt(2)*(f^5*cos(f*
x + e)^4 - 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*arctan(1/16*8^(3/4
)*sqrt(2)*(2*f^5*sqrt(f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(2)*f^2*sqrt(f^
(-4))*cos(f*x + e) - 8^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt
(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e
))/cos(f*x + e))*(f^(-4))^(3/4) - 1/8*8^(3/4)*(2*f^5*sqrt(f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-
4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + f^2*sqrt(f^(-4)) + sqrt(2))/f^4)/(f
*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{4}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(1+tan(f*x+e))**(3/2),x)

[Out]

Integral((tan(e + f*x) + 1)**(3/2)*cot(e + f*x)**4, x)

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Giac [A]
time = 1.56, size = 375, normalized size = 1.58 \begin {gather*} \frac {25 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{16 \, f} - \frac {25 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{16 \, f} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} + \frac {21 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 56 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} + 27 \, \sqrt {\tan \left (f x + e\right ) + 1}}{24 \, f \tan \left (f x + e\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

25/16*log(sqrt(tan(f*x + e) + 1) + 1)/f - 25/16*log(abs(sqrt(tan(f*x + e) + 1) - 1))/f + 1/2*(f^2*sqrt(2*sqrt(
2) + 2) - f*sqrt(2*sqrt(2) - 2)*abs(f))*arctan(1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) + 2*sqrt(tan(f*x + e) +
1))/sqrt(-sqrt(2) + 2))/f^3 + 1/2*(f^2*sqrt(2*sqrt(2) + 2) - f*sqrt(2*sqrt(2) - 2)*abs(f))*arctan(-1/2*2^(3/4)
*(2^(1/4)*sqrt(sqrt(2) + 2) - 2*sqrt(tan(f*x + e) + 1))/sqrt(-sqrt(2) + 2))/f^3 - 1/4*(f^2*sqrt(2*sqrt(2) - 2)
+ f*sqrt(2*sqrt(2) + 2)*abs(f))*log(2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e) + 1) + sqrt(2) + tan(f*x + e)
+ 1)/f^3 + 1/4*(f^2*sqrt(2*sqrt(2) - 2) + f*sqrt(2*sqrt(2) + 2)*abs(f))*log(-2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(t
an(f*x + e) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 + 1/24*(21*(tan(f*x + e) + 1)^(5/2) - 56*(tan(f*x + e) + 1)
^(3/2) + 27*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^3)

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Mupad [B]
time = 0.19, size = 173, normalized size = 0.73 \begin {gather*} -\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{8\,f}-\frac {\frac {9\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}-\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}+\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^4*(tan(e + f*x) + 1)^(3/2),x)

[Out]

atan(f*((1/2 - 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*1i)*((1/2 - 1i/2)/f^2)^(1/2)*2i - ((9*(tan(e + f*x) +
1)^(1/2))/8 - (7*(tan(e + f*x) + 1)^(3/2))/3 + (7*(tan(e + f*x) + 1)^(5/2))/8)/(f - 3*f*(tan(e + f*x) + 1) +
3*f*(tan(e + f*x) + 1)^2 - f*(tan(e + f*x) + 1)^3) - (atan((tan(e + f*x) + 1)^(1/2)*1i)*25i)/(8*f) + atan(f*((
1/2 + 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*1i)*((1/2 + 1i/2)/f^2)^(1/2)*2i

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